To let $ X $ to be a variety with one $ G $Action of an algebraic group on it.

My question refers to a motivating example

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here is the relevant excerpt:

Here the author discusses an example of $ X / G $ to explain that it is necessary to shape $ X / G $ as *categorical quotient* and not the *topologically* on.

We consider a motivational example introduced on page 27:

Here we take $ X: = mathbb {C} ^ 2 $ with action of $ G: = mathbb {C} ^ x $ by multiplication $ lambda cdot (x, y) mapsto ( lambda x, lambda y) $,

Obviously, the "naive" topological quotient theoretically consists of the lines $ {( lambda x, lambda y) vert lambda in mathbb {C} ^ x } $ and the origin $ {(0,0) } $,

Topologically, the origin lies in the closure of each line.

The QUESTION is why that argument already implies that $ Y: = X / G $ can not have a structure of a species? I do not understand the author's argument.

If we call with $ p: mathbb {C} ^ 2 to Y $ The canonical projection map and through (continuity?) This card can not separate orbits, which is why this implies $ Y $ has no structure of a sort as stated in the extract?

What role is particularly played by the fact that we can not separate the lines from the origin (in *purely topologically away*) It creates an obstacle to form a diverse structure $ X / G $?

Comment: I know that there are different ways to derive that when we define $ X / G $ then it can not have a topological structure. The most common argument is to introduce the invariant ring $ R ^ G $ and explicitly calculate here. But the main problem of this question is

It made me curious that the given reasoning seems a bit more "elemental" in the sense that in this example it does not explicitly work with the concept of the invariant ring $ R ^ G $,